Visibility phenomena in hypercubes

Abstract

We study the set of visible lattice points in multidimensional hypercubes. The problems we investigate mix together geometric, probabilistic and number theoretic tones. For example, we prove that almost all self-visible triangles with vertices in the lattice of points with integer coordinates in $\mathcal W=[0,N]^d$ are almost equilateral having all sides almost equal to $\sqrt{d}N/\sqrt{6}$, and the sine of the typical angle between rays from the visual spectra from the origin of $\mathcal W$ is, in the limit, equal to $\sqrt{7}/4$, as $d$ and $N/d$ tend to infinity. We also show that there exists an interesting number theoretic constant $\Lambda_{d,K}$, which is the limit probability of the chance that a $K$-polytope with vertices in the lattice $\mathcal W$ has all vertices visible from each other.